Biogeography-Based Optimization of PID Tuning Parameters for the Vibration Control of Active Suspension System

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1 CEAI, Vol.16, No.1 pp , 214 Printed in Romania Biogeography-Based Optimization of Tuning Parameters for the Vibration Control of Active Suspension System R. Kalaivani*, P. Lakshmi** *DEEE, CEG, Anna University, Guindy, Chennai -25 India ( lakshmi_ ). Abstract: This paper compares the Biogeography-Based Optimization (BBO) and Particle Swam Optimization (PSO) algorithms for the tuning of Proportional Integral Derivative () controller parameters and concludes BBO works well compared to other in reducing the body acceleration of Vehicle Active Suspension System (VASS). Biogeography a mushrooming nature enthused global optimization procedure, which is based on the study of the geographical distribution of biological organisms and a swarm intelligence technique are used to find the optimal parameters of the controller to improve the performance of VASS. Simulations of passive system, active system with controller with and without optimization are performed with dual bump, sinusoidal and random kind of road disturbances using MATLAB/SIMULINK. The simulation results indicate the improvement of results with the BBO algorithm. Keywords: vehicle suspension, vibration, controller, Biogeography-Based Optimization, Particle Swam Optimization. 1. INTRODUCTION The automobile industries promote the research on vibration control in order to guarantee the driving and travelling comfort to the passengers. Otherwise the vibration will lead to unwanted noise in the vehicle, damage to the fittings attached to the car and cause severe health problems such as increase in heart rate, spinal problems etc to the passengers. The suspension system of an automobile plays a vital role in vehicle handling and ride comfort. Handling of vehicle depends on the force acting between the road surface and the wheels. Ride comfort is related to vehicle motion sensed by the passenger. In order to improve the handling and ride comfort performance, instead of conventional static spring and damper system, semi-active and active systems are being developed. A semi-active suspension involves the use of dampers with variable gain. An active suspension involves the passive components augmented by actuators that supply additional forces. Alternatively, an active suspension system possesses the ability to reduce the acceleration of sprung mass continuously as well as to minimize suspension deflection which results in the improvement of tyre grip with the road surface (Seok_il Son, 1996; Hrovat D., 1997). Increased competition in the automotive market has forced industries to research on the control strategies of Vehicle Active Suspension System (VASS). In the past, many researchers discussed about a number of control approaches theoretically, simulated using simulation software, experimentally verified and proposed for the control of active suspension system. The survey of optimal control technique applications to the design of active suspensions are listed in (Hrovart D., 1993) and the emphasis is on Linear-Quadratic (LQ) optimal control. A methodology to design a controller with a model which includes the passenger dynamics is presented in (Esmailzadeh E. and Taghirad H. D., 1996). The comparison of Linear Matrix Inequality (LMI) based controller and optimal Proportional Integral Derivative () controller by (Abdalla MO et al., 27) proved the sprung mass displacement response improvement by LMI controller with only the suspension stroke as the feedback. controller for single Degree of Freedom (DOF), two DOF Quarter Car (QC) and half car model are discussed in (Abdalla MO et al., 27; Rajeswari K. and Lakshmi P., 211; Demir O. et al., 212) respectively. Design of robust PI controller which is used to obtain optimal control is discussed in (Yeroglu C. and Tan N. 28). The conclusion made by (Dan Simon, 28), stimulated the idea of using Biogeography-Based Optimization (BBO) for the optimization of parameters. (Panchal V. K. et al., 29) focused on classification of the satellite image of a particular land cover using the theory of BBO and concluded that with this highly accurate land cover features can be extracted effectively. (Urvinder Singh et al., 21) used BBO to optimize the element length and spacing for Yagi-Uda antenna and the performance is evaluated with a method of moment s code NEC2. (Aniruddha Bhattacharya and Pranab Kumar Chattopadhyay, 21) presented BBO algorithm to solve both convex and non-convex Economic Load Dispatch (ELD) problems of thermal plants and suggested that it is the promising alternative approach for solving the ELD problems in practical power system. (Dan Simon, 211) used Markov theory for partial immigration based BBO to derive a dynamic system model. (Abhishek Sinha et al., 211) given a better insight into the dynamics of

2 32 CONTROL ENGINEERING AND APPLIED INFORMATICS migration in actual biogeography systems and also helped in the understanding of the search mechanism of BBO on multimodal fitness landscapes. (Jamuna K. and Swarup K. S. 212) projected a multi-objective BBO algorithm to design optimal placement of phasor measurement units which makes the power system network completely observable and performed the simultaneous optimization of the two conflicting objectives such as minimization and maximization of two different parameters. (Aniruddha Bhattacharya and Pranab Kumar Chattopadhyay, 212) highlighted the effectiveness of BBO over Genetic Algorithm (GA) and Particle Swam Optimization (PSO) for an ELD problem. (Haiping Maa et al., 213) tested the performance of BBO for real-world optimization problems with some benchmark functions. (Lohokare M. R. et al., 213) presented an improved accuracy yielding algorithm in which the performance of BBO is accelerated with the help of a modified mutation and clear duplicate operators. Also discussed the suitability of BBO for real time applications. Tuning the controller parameters is a challenging task for the controller designers to achieve certain goals. Hence a choice is taken to optimize the tuning parameters of controller applied to VASS with BBO as it has the characteristics of information sharing among the solutions. To check the suitability of this optimization technique, another optimization method PSO which is developed by (Kennedy J. and Eberhart R. C., 1995) is used because it can generate a high quality solution with more unwavering convergence characteristics with less calculation time than other stochastic methods. Also it is attractive because there are only a few parameters to adjust. Compared to most of other evolutionary algorithms, in BBO and PSO each solution stay survive to the end of optimization procedure which enhances the capability of global search. The chance of the population convergence to the optimum is also more for BBO compared to GA. (Voratas Kachitvichyanukul, 212) highlighted the features PSO compared to GA. In (Zwe-Lee Gaing, 24), the author proposed PSO- controller for AVR system and shown that it is more efficient than the GA- controller. (Rajeswari K. and Lakshmi P. 21b) optimized a Fuzzy Logic Controller (FLC) used in VASS with two optimization algorithms GA and PSO and concluded that PSO tuned FLC based active suspension system exhibits an improved ride comfort and good road holding ability. (Shen-Lung Tung et al., 211) proposed an active suspension mechanism for three DOF twin-shaft vehicles of front axle suspension with bounded uncertainties using exponential decay control and PSO techniques. (Juing- Shian Chiou et al., 212) presented a design method for determining the optimal fuzzy controller parameters of active automobile suspension system using PSO. (Oscar Castillo and Patricia Melin, 212a) considered the application of GA, PSO and Ant Colony Optimization (ACO) as three different paradigms that help in the design of optimal type-2 fuzzy controllers and concluded that both PSO and ACO are able to outperform GA. Wu Q H and Liao H L (213) used PSO for comparison with newly presented algorithm, Function Optimization by Learning Automata (FOLA), to solve complex function optimization problems. In this paper, an attempt is made to optimize the controller parameters with two different optimization techniques namely PSO and BBO and the comparison is made with the outputs of linear system model which is considered for the simulation study. Organization of the paper is as follows. QC model dynamics of an active suspension system is briefly explained in section 2. Discussion of control scheme is presented in section 3. PSO and PSO based () are discussed in sections 4. In section 5, BBO and BBO based () are discussed. In section 6, the simulation results are presented and discussed. The final section concludes the paper. 2. ACTIVE SUSPENSION QUARTER CAR MODEL A two DOF QC model of VASS is shown in Fig. 1. It represents the automotive system at each wheel i.e. the motion of the axle and the vehicle body at any one of the four wheels of the vehicle. QC model used in (Rajeswari K. and Lakshmi P., 21a) is considered because it is simple and one can observe the basic features of the VASS such as sprung mass displacement, body acceleration, suspension deflection and tyre deflection. ks m s m u Fig. 1. Two DOF QC model. bs The suspension model consists of a spring k s, a damper b s and an actuator of active force F a. For a passive suspension, Fa can be set to zero. The sprung mass m s represents the QC equivalent of the vehicle body mass. An unsprung mass m u represents the equivalent mass due to axle and tyre. The vertical stiffness of the tyre is represented by the spring k t. The variables z s, z u and z r represents the vertical displacements from static equilibrium of sprung mass, unsprung mass and the road respectively. Equations of motion of two DOF QC model of VASS are given in (1). It is assumed that the suspension spring stiffness and tyre stiffness are linear in their operating ranges and that the tyre does not leave the ground. The state space representation of QC model is given in (2). k t F a z r z s z u

3 CONTROL ENGINEERING AND APPLIED INFORMATICS 33 ms z s+ b s (z s - z u ) + k s (z s - z u ) = Fa mu z u - b s (z s - z u ) - k s (z s - z u ) - k t (z r - z u ) = - Fa (1) 1 X X 1 -ks -bs ks bs 1 1 X 2 s s s s X m m m m 2 m s a X 3 1 X3 kt X 4 ks bs -ks -kt -bs X4-1 mu mu mu mu m u m u = + F + z X = AX + BF + Fz (3) a where X 1= z s, X 2= z s, X 3= z u, X 4= zu r The natural frequency of unsprung mass is ω = k+k s t m u 3. CONTROL STRUCTURE controller (Rajeswari K. and Lakshmi P. 211) which is a combination of proportional, integral and derivative controller can improve the total performance of the system. In other words, both the transient and steady state response can be improved. r (2) (4) To reduce the effect of road disturbance input (Fig. 2), two QC suspension parameters such as suspension deflection ( z s -z u ) and sprung mass velocity ( z s ) are feedback to the controllers (Jyh-Chyang Renn, and Tsung-Han Wu 27). The feedback gains are Ge and G v respectively. The output control signals are amplified by a gain G u and then given as the input to the actuator. In this work the nonlinear dynamics of actuator is not considered and the gain of the linear actuator is taken as G a. The actuator force F a which is the additional input to the system is proportional to the controller output to have better comfort. Controllers are tuned by three methods -robust response tuning, PSO tuning and BBO tuning. First, autotuning has been carried out for both 1 and 2 to give robust performance using the MATLAB simulation software. Next, for optimized tuning, the RMS value of body acceleration ( z s ) is taken as the performance index. The objective is to find tuning parameters of 1 ((k p1, k i1 & k d1 ) and 2 (k p2, k i2 & k d2 ) to minimize the cumulative RMS value of body acceleration signal ( z s ). In other words, 1 2 Minimize J (K)= zs dt T (6) where T is the total time period T 4. PARTICLE SWAM OPTIMIZATION Fig. 2. The block diagram representation of control scheme of VASS using controller. de (5) U c= k pe(t) + ki e(t)dt + kd dt where 1 + G e U c is the controller output Road disturbance Fa QC Model 2 G u G a + Sprung mass velocity G v Suspension deflection Start Generate an initial population Calculate the fitness value No satisfied PSO update velocity PSO- update global best position Yes Stop PSO update position PSO- update individual best velocity k p k i k d e(t) Proportional gain Integral gain Differential gain Input to the controller e(t)dt Time integral of the input signal de Time derivative of the input signal dt Perform Mutation Fig. 3. The Flow Chart for PSO Optimization. Perform Reproduction A population based stochastic optimization technique PSO which is developed because of the insight of group actions of bird flocking or fish schooling is having immense curiosity among the optimizing techniques In PSO, the optimization procedure starts with the initialization of the population of

4 34 CONTROL ENGINEERING AND APPLIED INFORMATICS random solutions as in GA and searches for optima by updating generations (Oscar Castillo and Patricia Melin, 212b). Compared to other evolutionary algorithms, in PSO the individual solutions called particles, fly through the problem space by following the current optimum particles with the change of its velocity and position. Each and every particle keeps the pathway of its coordinates in the problem space, which are associated with the best solution it has achieved so far in each generation. In every iteration, each particle is updated by two "best" values Pbest and Gbest. Out of which the Pbest is the best solution (fitness) it has achieved so far and the Gbest is the best solution obtained so far by any particle in the population. Each particle knows Pbest and Gbest. Each particle tries to modify its position using the current velocity and the distance from Pbest and Gbest. The modified velocity and position of each particle can be calculated using equations given below. v = w* v +c * rand *(Pbest - x )+c * rand *(Gbest - x ) (7) k+1 k k k i,g i,g 1 1 i,g i,g 2 2 g i,g k+1 k k+1 x i,g = x i,g + v i,g (8) i =1, 2,..n g =1, 2,..m where n - Number of particles in a group m - Number of members in a particle c, 1 c 2 - Weight factors k v i, g - Velocity of the particle i in k th iteration k x i, g - Position of particle i in k th iteration rand 1, rand 2 - Random numbers between and 1 Pbest i,g - Pbest of particle i Gbest g -Gbest of the group Proper selection of inertia weight parameter w by (9) provides a balance between global and local explorations. t w = w max -((w max - w min )* ) t where t max w max and w min are the upper and lower bounds for w. - Current iteration number t max - Maximum number of iterations The flowchart for optimization steps of PSO algorithm (Juing-Shian Chiou et al. (212)) is shown in Fig controller In this paper, PSO algorithm is used to find the parameters (Zwe-Lee Gaing, 24) of both the controllers [ k p1, k i1, k d1, k p2, k i2, k d2 ] which represents an individual K. Hence there are six members in a paricle (m). Since the number of individuals in a group is n, the population dimension is n x 6. The fitness function f is the performance index J (K). f = J(K) (1) (9) For each variables to be tuned the velocity is updated as in (7), the position is updated as in (8) The optimization steps explained in section 4 are followed by maintaining the values with in the lower and upper bounds. 5. BIOGEOGRAPHY-BASED OPTIMIZATION Another population based BBO technique has been developed based on the theory of Biogeography (Yeroglu C. and Tan N., 28) which describes how species voyage from one habitat to other, how new species come up and how species become vanished. A habitat is a geographically isolated island from other habitats. Each habitat has its individual features which are specified by the Habitat Suitability Index (HSI) variables. A habitat with high HSI is well suited for species living. The migration of a species among habitats takes place when the high HSI habitats have more species or habitat has low HSI. This process is known as emigration. Another process called immigration takes place when the species move towards the habitat with high HSI having few species. The emigration and immigration of species from a habitat are called migration. The emigration rate (µ) and immigration rate (λ) vary with the number of species available in the habitat. With no species the immigration rate touches the upper limit and with maximum number of species it is zero where as the emigration rate increases with increase in the number of species. The change in HSI due to natural disaster is taken into account with the mutation operation. The optimization steps of BBO algorithm (Jamuna K. and Swarup K. S., 212) are described as follows: 1. Initialize the optimization problem and the BBO parameters. 2. Initialize randomly the habitat variables. 3. Perform BBO migration operation as in definition 7 and mutation operation as in definition 8 (Yeroglu C. and Tan N., 28) and compute the HSI. 4. The emigration and immigration rates of each solution are useful in probabilistically sharing the information between the habitats. Each solution can be modified with the habitat modification probability to yield good solution. Recompute HSI and modify the habitats. 5. Check the stopping criteria. If not achieved, repeat from step3. BBO does not involve the reproduction of solution as in GA. In each generation, the fitness of every solution (habitat) is used to find the migration rates controller As discussed in section 4.1, to find the individual K, instead of PSO, BBO is introduced. With the initial random choice of the individual K, depending on the emigration rate and immigration rates decided by the HSI allows the survival of a habitat in the search space. Good solutions can be achieved by increasing the diversity of population by mutation operation. Mapping of HSI to the maximum number of species S max, calculation of the emigration rate and

5 CONTROL ENGINEERING AND APPLIED INFORMATICS 35 immigration rate using equations 11 and 12 are done. d λ = 1(1 - ) (11) S max max E d μ = (12) S where d is the number of species at the instant of time. Final values of individual K are taken by checking the fitness function. 6. SIMULATION The parameters of the quarter car model taken from Rajeswari K and Lakshmi P (21a) are listed below. Sprung mass ( m s ) = 29 kg Unsprung mass ( m u ) = 59 kg Damper coefficient ( b s ) = 1, Ns/m Suspension stiffness ( k s ) = 16,812 N/m Tyre stiffness ( k t ) = 19, N/m Bump Height (m) Sin Input (m) Random Input (m) Bump Input Fig. 4. Road Input profile Dual Bump Input Sinusoidal Input Random input. International Organization for Standardization gives the classification of road roughness using Power Spectral Density values. In this work three kinds of the road disturbance are considered initially a dual bump and for robustness checking a sinusoidal and random input. Mathematical representation of the dual bump input with 1cm and 5cm amplitude can be stated as a 1(-cos(8πt)) z(t) r = if 1. t a 2(-cos(8πt)) if 3. t otherwise (13) where a 1 and a 2 denotes the two bump amplitudes. The following parameters are initialized for PSO. Population size : 1 Number of variables to be tuned : 6 Initial population : Random selection Initial velocity : Mutation rate :.5 Number of iterations : 1 Weight factors (c 1, c 2 ) :.5,.5 Minimum and maximum parameter values of decision variable (K) : Table 1 The following parameters are initialized for BBO. Population size : 1 The maximum species count (S max ) : 5 Migration probability : 1 Mutation probability :.5 Selectiveness parameter δ : 2 Max migration rate I : 1 Max emigration rate E : 1 Step size used for numerical integration : 1 Minimum and maximum parameter values of decision variable (K) : Table 1 Lower bound and Upper bound for immigration probability per gene : [.1, 1] Table 1. Range of the controller tuning parameters. Tuning Parameter Lower Bound Upper Bound k p1.1 k i1 1 k d1.5 k p2 1 k i2 1 k d2 1 Objective function value (x1-3 m/s 2 ) Iterations Fig. 5. Statistics of search process. PSO BBO

6 36 CONTROL ENGINEERING AND APPLIED INFORMATICS The optimization results were computed by averaging 5 minimization runs and the convergence characteristics of each technique are shown in Fig. 5. Each run yielded the global minimum results. From the convergence plot, BBO algorithm is found to be superior to PSO algorithm discussed. Also the dynamic behaviours and convergence characteristics of the algorithms can be analyzed with the statistical indices mean (M) and standard deviation (σ) which are given by M = σ = n i=1 1 f(k ) n i (14) n 2 (f(k i )-M) (15) n i=1 where population size. f(k i ) the fitness value of the individual Ki and n is the The BBO algorithm results better fitness value and mean value compared PSO algorithm (Table 2). Table 2. Comparison of computational efficiency of PSO and BBO algorithms. Algorithm Maximum Minimum Range M σ PSO BBO The robust response tuning, PSO and BBO optimized values of tuning parameters which are obtained are appended to Table 3. Table 3. The Optimized tuning parameters with PSO and BBO. Controller Parameter Robust response tuning kp ki kd x kp ki kd The mathematical model of vehicle suspension system (1) with the and controllers discussed in sections 4 and 5 are simulated with all the mentioned road input profiles (Fig. 4). The simulation results of passive system, system with, and controllers are shown in Fig. 6, 7 and 8. - (d). Also the RMS values of the time responses of the system are tabulated in Tables 4-6. Sprung M ass (m) Body Acceleration (m /s 2 ) Suspension Deflection (m ) Tyre Deflection (m) (d) Fig. 6. Time responses with dual bump input Sprung mass displacement Body acceleration Suspension deflection and (d) Tyre displacement.

7 CONTROL ENGINEERING AND APPLIED INFORMATICS 37 Sprung Mass (m) Sprung Mass displacement (m) Body Acceleration (m/s 2 ) Suspension Deflection (m ) Tyre deflection (m) (d) Fig. 7. Time responses with sinusoidal input Sprung mass displacement Body acceleration Suspension deflection and (d) Tyre displacement. Body Acceleration (m/s 2 ) Suspension deflection (m) Tyre Deflection (m) (d) Fig. 8. Time responses with random input Sprung mass displacement Body acceleration Suspension deflection and (d) Tyre displacement. It is clear from Fig. 6, 7, 8. and Fig. 6, 7, 8. that the vehicle body acceleration and sprung mass displacement are considerably reduced by the proposed controller. It guarantees the travelling comfort to the passengers. Both the designed and have robust performance which is true from Fig. 7, 8.. Also Fig. 6, 7, 8. shows that the suspension deflection by both the controllers is nearly the same. Fig. 6, 7, 8. (d) illustrates the road holding ability maintained by both the controllers. Tyre displacement of active systems is higher than that of the passive suspension system. Power Spectral density (m 2 sec -4 /Hz) Frequency (Hz)

8 38 CONTROL ENGINEERING AND APPLIED INFORMATICS Power Spectral Density (m 2 s -4 /Hz) Pow er Spectral D ensity (m 2 s -4 /Hz) Frequency (Hz) Frequency (Hz) Fig. 9. PSD of Body acceleration comparison of, and with a) Dual bump b) Sinusoidal c) Random inputs. Table 4. RMS values of the time responses of quarter car model with dual bump input. System Sprung Mass x1-3 (m) Body Acceleration x1-3 (m/s 2 ) Suspension Deflection x1-3 (m) Tyre (m) Table 5. RMS values of the time responses of quarter car model with sinusoidal input. System Sprung Mass x1-3 (m) Body Acceleration x1-3 (m/s 2 ) Suspension Deflection x1-3 (m) Tyre (m) Table 6. RMS values of the time responses of quarter car model with random input. System Sprung Mass x1-3 (m) Body Acceleration x1-3 (m/s 2 ) Suspension Deflection x1-3 (m) Tyre (m) The RMS values of the time responses of the four outputs with different inputs are listed in Table 4, 5 and 6. It is clear that the VASS using controller is useful for betterment of ride and travelling comfort with reduced body acceleration over controller and passive system. In the evaluation of vehicle ride quality, the Power Spectral Density (PSD) for the body acceleration as a function of frequency is of prime interest and is plotted for passive and system with and controller for three different types of road inputs (Fig. 9). It is clear from the PSD plot that in the human sensitive frequency range 4-8 Hz, compared to and, reduces the vertical vibrations to a great extent and improves the comfort of travelling. 8. CONCLUSIONS In this paper the PSO and BBO optimization of tuning parameters has been discussed for application in QC model with linear actuator. Repeated runs of both the optimization techniques have been carried out 5 times and the simulations are carried out with most fitted values. Among the two optimization techniques, the BBO gives better performance. With controller, the results are improved compared to passive system, conventional and controllers for the control of VASS. reduces the body acceleration considerably and ensures the travelling comfort to the passengers. The controllers discussed are easy to implement and with reference to the PSD of body acceleration, it is clear that all the controllers provides better vibration control compared to the passive system. gives better PSD and proves its effectiveness for the control of vibration in comparison with the other two controllers discussed. In future, the BBO can be hybridized to increase the convergence speed.

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